JP7526367B2 - Motional mode configurations for the implementation of entanglement gates in trapped-ion quantum computers - Google Patents

Description

The present disclosure relates generally to methods for generating entanglement gates in trapped ion quantum computers, and more specifically to methods for constructing simple laser pulses to generate entanglement gates using a kinetic mode structure configured such that the pulses can be practically implemented.

In quantum computing, quantum bits, or qubits, similar to the bits that represent "0" and "1" in classical (digital) computers, must be prepared, manipulated, and measured (read out) with near-perfect control during the computational process. Imperfect control of qubits can allow errors to accumulate in the computational process, limiting the size of quantum computers that can perform reliable calculations.

Among the physical systems proposed for building large-scale quantum computers are chains of ions (e.g., charged atoms) trapped by electromagnetic fields and suspended in a vacuum. The ions have internal hyperfine states that are separated by frequencies in the several GHz range and can be used as computational states of qubits (called "qubit states"). These hyperfine states can be controlled using radiation provided by a laser, or sometimes referred to herein as interaction with a laser beam. Ions can be cooled to near their motional ground state using such laser interactions. Ions can also be optically excited with high precision into one of two hyperfine states (qubit preparation), manipulated between the two hyperfine states by a laser beam (single-qubit gating), and their internal hyperfine states detected by fluorescence upon application of a resonant laser beam (qubit readout). A pair of ions can be controllably entangled by a force that depends on the qubit state, using a laser pulse that couples the ions into a collective motional mode of the chain of trapped ions, arising from the interaction of Coulomb forces between the ions (two-qubit gating).

However, trains of laser pulses to perform computational processes in large-scale quantum computing introduce technical complexities in the optics and electronics to modulate such laser pulses. Therefore, a procedure is needed to reduce the complexity in the physical system to control qubits to perform the desired computational processes.

Embodiments of the present disclosure provide a method of performing a computation using a quantum computer. The method includes modulating a motional mode structure of a plurality of trapped ions, each of which has two frequency-separated states that define a qubit; calculating a detuning frequency function and an amplitude function of a laser pulse to induce entanglement interactions between pairs of trapped ions of the plurality of trapped ions; and performing a quantum computation in the quantum computer by applying a laser pulse having the calculated detuning frequency function and amplitude function to the pairs of trapped ions for a gate duration.

Embodiments of the present disclosure also provide a non-volatile computer readable medium including computer program instructions. The computer program instructions, when executed by a processor, cause the processor to calculate a detuning frequency function and an amplitude function of a laser pulse to induce entanglement interactions between pairs of trapped ions of a plurality of trapped ions, and perform a quantum computation in a quantum computer by applying a laser pulse having the calculated detuning frequency function and amplitude function to the pairs of trapped ions for a gate duration. The motional mode frequencies of the plurality of trapped ions are modulated to be integer multiples of 4π divided by the gate duration.

Embodiments of the present disclosure further provide a quantum computing system. The quantum computing system includes a plurality of trapped ions in an ion trap, each of the trapped ions having two hyperfine states that define a qubit, and a controller having a non-volatile memory having a number of instructions stored therein, which, when executed by the processor, cause the quantum computing system to perform operations including: calculating a detuning frequency function and an amplitude function of a laser pulse to cause entanglement interactions between pairs of trapped ions of the plurality of trapped ions; and performing a quantum computation in a quantum computer by applying a laser pulse having the calculated detuning frequency function and amplitude function to the pairs of trapped ions for a gate duration. The confinement potential of the ion trap is modulated such that the motional mode frequencies of the plurality of trapped ions are integer multiples of 4π divided by the gate duration.

So that the above features of the present disclosure may be understood in detail, a more particular description of the present disclosure briefly summarized above can be made by reference to embodiments, some of which are illustrated in the accompanying drawings. It should be noted, however, that the accompanying drawings illustrate only exemplary embodiments of the present disclosure and should not be considered as limiting its scope, since the present disclosure may admit of other equally effective embodiments.

FIG. 1 is a partial view of a trapped ion quantum computer according to one embodiment. 1 shows a schematic diagram of an ion trap for confining ions in chains according to one embodiment. 3A, 3B, and 3C show some schematic collective transverse motion mode structures of a chain of five trapped ions. 1 shows a schematic energy diagram of each ion in a chain of trapped ions according to one embodiment. 1 shows the qubit states of the ions represented as points on the surface of the Bloch sphere. 1 shows a schematic diagram of the motional sideband spectrum and motional modes of each ion according to one embodiment. 1 shows a schematic diagram of the motional sideband spectrum and motional modes of each ion according to one embodiment. 1 shows a numerically calculated profile of the static (DC) voltage _Vs applied to trapping electrodes according to one embodiment. 1 shows the numerically calculated gate power ratios |χ _ij /Ω ² τ ² | according to one embodiment. 4 shows a numerically calculated pulse function according to one embodiment. 4 shows a numerically calculated pulse function according to one embodiment. 1 shows the numerically calculated residual coupling α versus variation in modal frequency δω, according to one embodiment. 11 shows a flow chart illustrating a method 1100 used to perform a calculation using a quantum computer.

For ease of understanding, the same reference numbers are used, where possible, to designate identical elements common to the figures. The figures and the following description use a Cartesian coordinate system including an X-axis, a Y-axis, and a Z-axis. Directions represented by arrows in the figures are assumed to be positive for convenience. It is believed that elements disclosed in some embodiments may be beneficially utilized in other implementations without specific specification.

The embodiments described herein generally relate to methods and systems for constructing and delivering pulses for performing entanglement gate operations between two trapped ions during quantum computation, and more specifically, to methods for designing pulses that can be practically implemented in a system while simplifying the system complexity and achieving high fidelity of the entanglement gate operation, or high probability that at least two ions are in the intended qubit state after performing the entanglement gate operation between the two ions. It should be noted that while a method for entanglement gate operations between two ions is described herein, the method can also be used for single qubit operations and entanglement operations between more than two ions.

The overall system capable of performing quantum computations using trapped ions includes a classical computer, a system controller, and a quantum register. The classical computer performs support and system control tasks including selecting a quantum algorithm to execute using a user interface such as a graphics processing unit (GPU), compiling the selected quantum algorithm into a series of universal logic gates, translating the series of universal quantum logic gates into laser pulses for application to the quantum register, and using a central processing unit (CPU) to pre-calculate parameters to optimize the laser pulses. A software program for performing the tasks of decomposing and executing the quantum algorithm is stored in a non-volatile memory within the classical computer. The quantum register includes trapped ions coupled with various hardware, including a laser that manipulates the internal hyperfine state (qubit state) of the trapped ions, and an acousto-optic modulator that reads out the internal hyperfine state (qubit state) of the trapped ions. The system controller receives the pre-calculated parameters of the pulses from the classical computer at the start of execution of the algorithm selected in the quantum register, controls the various hardware associated with controlling any and all aspects used to execute the algorithm selected in the quantum register, and returns the readout of the quantum register, thus outputting the results of the quantum computation to the classical computer at the end of execution of the algorithm.

The methods and systems described herein also include processes for converting logic gates into laser pulses that are applied to the quantum registers, and processes for pre-calculating parameters that optimize the laser pulses that are applied to the quantum registers and used to improve the performance of the quantum computer.

Among several known sets of universal logic gates into which any quantum algorithm can be decomposed, a set of universal logic gates, generally denoted as {R, XX}, is unique to the trapped ion quantum computing system described herein. Here, the R gate corresponds to the manipulation of the individual qubit states of the trapped ion, and the XX gate (also called the "entanglement gate") corresponds to the manipulation of the entanglement of two trapped ions. As will be apparent to those skilled in the art, the R gate can be implemented with near perfect fidelity, but the formation of the XX gate is complex, and thus, in order to improve the fidelity of the XX gate and avoid or reduce errors in the computations in the quantum computer, optimization of the given type of trapped ion, the number of ions in the chain of trapped ions, and the hardware and environment in which the trapped ions are trapped, to name a few factors, is required. The following discussion describes how to generate and optimize the pulses used to perform computations based on the formation of the XX gate with improved fidelity.

As quantum computers grow in size, the entanglement gate operations used to perform quantum computations become more and more complex, and the pulses used to perform these entanglement gate operations also become more and more complex. This increased complexity can create practical limitations in implementing the pulses. The methods and systems described in this disclosure modify such pulses so that they can be practically implemented without sacrificing precise control of the qubits.

General Hardware Configuration Figure 1 is a partial diagram of a trapped ion quantum computer or system 100 according to one embodiment. The system 100 includes a classical (digital) computer 101, a system controller 118, and a quantum register that is a chain of trapped ions (e.g., five are shown) 102 extending along the Z-axis. Each ion in the chain of trapped ions 102 is an ion having a nuclear spin I and an electron spin S such that the difference between the nuclear spin I and the electron spin S is zero, e.g., a positive ytterbium ion ¹⁷¹ Yb ⁺ , a positive barium ion ¹³³ Ba ⁺ , a positive calcium ion ¹¹¹ Cd ⁺ or ¹¹³ Cd ⁺ , all of which have a nuclear spin I = 1/2 and ^{a 2} S _1/2 hyperfine state. In some embodiments, all ions in the chain of trapped ions 102 are of the same species and isotope (e.g., ¹⁷¹ Yb ⁺ ). In some other embodiments, the chains of trapped ions 102 include more than one species or isotopes (e.g., some ions are ¹⁷¹ Yb ⁺ and some other ions are ¹³³ Ba ⁺ ). In still further embodiments, the chains of trapped ions 102 may include various isotopes of the same species (e.g., different isotopes of Yb, different isotopes of Ba). The ions in the chains of trapped ions 102 are individually treated with separate laser beams.

Classical computer 101 includes a central processing unit (CPU), memory, and support circuits (or I/O). The memory is connected to the CPU and may be one or more of a readily available memory, such as read-only memory (ROM), random access memory (RAM), floppy disk, hard disk, or any other form of digital storage, either locally or remotely. Software instructions, algorithms, and data may be coded and stored in the memory to instruct the CPU. Support circuits (not shown) are also connected to the CPU to support the processor in a conventional manner. The support circuits may include conventional cache, power supplies, clock circuits, input/output circuits, subsystems, and the like.

An imaging objective 104, for example a 0.37 numerical aperture (NA) objective, collects fluorescence from the ions along the Y axis and maps each ion to a multichannel photomultiplier tube (PMT) 106 for measuring individual ions. A non-copropagating Raman laser beam from a laser 108, provided along the X axis, performs operations on the ions. A diffractive beam splitter 110 creates an array of static Raman beams 112 that are individually switched using a multichannel acousto-optic modulator (AOM) 114 and are configured to selectively act on individual ions. A global Raman laser beam 116 illuminates an ion at a time. In some embodiments, each of the individual Raman laser beams (not shown) illuminates an individual ion. A system controller (also called an "RF controller") 118 controls the AOM 114. The system controller 118 includes a central processing unit (CPU) 120, a read only memory (ROM) 122, a random access memory (RAM) 124, a storage unit 126, and the like. The CPU 120 is the processor of the RF controller 118. The ROM 122 stores various programs, and the RAM 124 is a working memory for various programs and data. The storage unit 126 includes a non-volatile memory such as a hard disk drive (HDD) or a flash memory, and stores various programs even when the power is turned off. The CPU 120, the ROM 122, the RAM 124, and the storage unit 126 are interconnected via a bus 128. The RF controller 118 executes a control program that is stored in the ROM 122 or the storage unit 126 and uses the RAM 124 as a working area. The control program includes one or more software applications containing program code (e.g., instructions) that can be executed by the processor to perform various functions related to receiving and analyzing data and controlling any and all aspects of the methods and hardware used to create the trapped ion quantum computer system 100 described herein.

FIG. 2 shows a schematic diagram of an ion trap 200 (also called a Paul trap) that confines ions in the chains 102, according to one embodiment. The confinement potential is applied by both static (DC) and radio frequency (RF) voltages. A static (DC) voltage V _S is applied to end cap electrodes (also called "DC control electrodes") 210 and 212 to confine ions along the Z-axis (also called the "axial", "longitudinal" or "first direction"). The ions in the chains 102 are approximately evenly distributed in the axial direction due to Coulomb interactions between the ions. In some embodiments, the ion trap 200 includes four hyperbolic shaped electrodes (also called "RF rails") 202, 204, 206, and 208 that extend along the Z-axis.

In operation, a sinusoidal voltage _V1 (with amplitude _VRF /2) is applied to one pair of opposing electrodes 202, 204, and a sinusoidal voltage V2 with a phase shift of 180° from the sinusoidal voltage _V1 (and amplitude _VRF / ₂ ) is applied to the other pair of opposing electrodes 206, 208 at a driving frequency _ωRF to generate a quadrupole potential. In some embodiments, the sinusoidal voltage is applied only to the opposing pair of electrodes 202, 204, and the other pair of opposing electrodes 206, 208 is grounded. The quadrupole potential generates an effective confining force in the XY plane perpendicular to the Z axis (also called the "radial", "lateral" or "second direction") for each trapped ion, which is proportional to the distance from the saddle point (i.e., the location in the axial (Z) direction) where the RF field vanishes. The radial (i.e., in the XY plane) motion of each ion can be approximated as a harmonic oscillation (called "secular motion") with a restoring force towards a radial saddle point, modeled by spring constants _kx and _ky , respectively, as described in more detail below. In some embodiments, the radial spring constants are modeled as being equal when the quadrupole potential is radially symmetric. However, if undesirable, the radial ion motion may be distorted due to some asymmetry in the physical trapping configuration, small DC patch potentials due to non-uniformities on the surfaces of the electrodes, etc., and these and other external sources of distortion may cause the ion to be de-centered from the saddle point.

It should be noted that the specific example of a conventional macroscopic ion trap shown in FIG. 2 is merely a possible example of an ion trap for trapping ions and is not intended to limit the possible configurations, specifications, etc. of the ion trap according to the present disclosure. For example, the ion trap may be a microfabricated surface trap in which the electromagnetic confinement potential required to trap ions is formed on the surface of a semiconductor chip. Examples of such surface traps include the Sandia high-optical access (HOA) 2.0 trap and the GTRI/Honeywell ball-grid array (BGA) trap, both of which are known in the art. Such microfabricated surface traps are known to offer superior ability to design the geometry of the ion trap and configure various parameters of the electromagnetic confinement potential with high reproducibility and manufacturing yield compared to manually assembled macroscopic ion traps.

3A, 3B, and 3C show some schematic collective transverse motion mode configurations (also referred to simply as "motion mode configurations") of, for example, five strands of trapped ions 102, where the confining potential due to the static voltage V _S applied to the end cap electrodes 210 and 212 is weak compared to the radial confining potential. The transverse collective motion mode of the strands of trapped ions 102 is determined by a combination of the confining potential generated by the ion trap 200 and the Coulombic interactions between the trapped ions. The trapped ions undergo collective transverse motion (referred to as "collective transverse motion modes", "collective motion modes", or simply "motion modes"), with each mode having a different energy (or equivalently, frequency) associated with it. In the following, the p-th lowest energy motional mode (or equivalently, mode frequency ω _p ) is referred to as |n _ph > _p , where n _ph represents the number of motion quanta (unit of energy excitation, called "phonon") of the motional mode, and the number P of a given transverse motional mode is equal to the number N of trapped ions in the chain 102. Figures 3A-3C illustrate schematic examples of different types of collective transverse motional modes that may be experienced by five trapped ions arranged in the chain 102. Figure 3A is a schematic diagram of the general motional mode (also called "center of mass (COM) mode") |n _ph > _P with the highest energy, where P is both the number of modes and the total number of motional modes. In the general motional mode | _{n ph} > _p , all ions oscillate in phase in the transverse direction. Figure 3B is a schematic diagram of the tilted motional mode (simply called the "tilt mode") |n _ph > _P-1 with the second highest energy. In the tilted motion mode, the ions at both ends move laterally out of phase (i.e., in opposite directions). Figure 3C is a schematic diagram of a higher-order motion mode (also called a "zigzag mode") |n _ph > P _-3, which is lower in energy than the tilted motion mode |n _ph > _P-1 and in which ions move in a more complex mode pattern.

Note that the above specific configuration is merely one of several possible examples of a trap that confines ions according to the present disclosure, and does not limit the possible configurations, specifications, etc. of the trap according to the present disclosure. For example, the shape of the electrodes is not limited to the above hyperbolic electrodes. In other examples, the trap that generates an effective electric field that causes the motion of ions in a radial direction as a harmonic oscillation may be a multi-layer trap in which multiple electrode layers are stacked and an RF voltage is applied to two diagonal electrodes, or it may be a surface trap in which all electrodes are arranged in a single plane on the chip. Furthermore, the trap can be divided into several segments, adjacent pairs of which can be linked to shuttle one or more ions or can be coupled by photonic interconnects. The trap can also be an array of individual trapping regions arranged close to each other on a microfabricated ion trap chip. In some embodiments, the quadrupole potential has a spatially varying DC component in addition to the above RF component.

4 shows a schematic energy diagram 400 of each ion in the chain of trapped ions 102 according to one embodiment. Each ion in the chain of trapped ions 102 is an ion with a nuclear spin I and an electronic spin S such that the difference between the nuclear spin I and the electronic spin S is zero. In one example, each ion may be a positive ytterbium ion ₁₇₁ Yb + with a nuclear spin I = 1/2 and ^{a 2} S _1/2 hyperfine state (i.e., two electronic states) with an energy splitting corresponding to a frequency difference (referred to as the "carrier frequency") of ω 01 /2π = 12.642812 ^GHz . In another example, each ion may be a positive barium ion ¹³³ Ba ⁺ , a positive cadmium ion ¹¹¹ Cd ⁺ or ¹¹³ Cd ⁺ , all with a nuclear spin I = ^1/2 and a ² S _1/2 hyperfine state. The qubit is formed in two hyperfine states, denoted |0〉 and |1〉, and the hyperfine ground state (i.e., the lower energy state of the 2S1 _/ ² hyperfine states) is selected to represent |0〉. Hereinafter, the terms "hyperfine state", "internal hyperfine state" and "qubit" may be used interchangeably to represent |0〉 and |1〉. Each ion can be cooled (i.e., the kinetic energy of the ion can be lowered) to near the motional ground state |0〉 _p of any motional mode p without phonon excitation (i.e., n _ph =0) with known laser cooling methods, such as Doppler cooling or resolved sideband cooling, and then the qubit state can be prepared in the hyperfine ground state |0〉 by optical pumping. Here, |0〉 represents an individual qubit state of the trapped ion, and |0〉 _p with the subscript p represents the motional ground state of motional mode p of the chain of trapped ions 102.

The individual qubit states of each trapped ion can be manipulated, for example, by a 355 nanometer (nm) mode-locked laser via the excited ² P _1/2 level (represented by |e〉). As shown in FIG. 4, the laser beam from the laser can be split into a pair of non-copropagating laser beams (the first laser beam with frequency ω ₁ and the second laser beam with frequency ω ₂ ) in a Raman configuration and detuned by a one-photon transition detuning frequency Δ=ω ₁ −ω _0e with respect to the transition frequency ω _0e between |0〉 and |e〉 as explained in FIG. 4. The two-photon transition detuning frequency δ involves tuning the amount of energy provided to the trapped ion by the first and second laser beams, which, when used in combination, move the trapped ion between hyperfine states |0〉 and |1〉. When the one-photon transition detuning frequency Δ is much larger than the two-photon transition detuning frequency (also simply called the “detuning frequency”) δ = ω ₁ - ω ₂ - ω ₀₁ (hereafter denoted as ±μ, where μ is a positive value), Rabi flopping occurs between states |0〉 and |e〉, and between states |1〉 and |e〉, respectively, at single-photon Rabi frequencies Ω _0e (t) and Ω _1e (t) (which are time-dependent and determined by the amplitudes and phases of the first and second laser beams), as well as the spontaneous emission rate from the excited state |e〉. Rabi oscillations between the two hyperfine states |0〉 and |1〉 (called “carrier transitions”) are induced at the two-photon Rabi frequency Ω(t). The two-photon Rabi frequency Ω(t) has an intensity (i.e., absolute value of amplitude) proportional to Ω _0e Ω _1e /2Δ, where Ω _0e and Ω _1e are the single-photon Rabi frequencies due to the first and second laser beams, respectively. Hereinafter, this set of non-copropagating laser beams in a Raman configuration for manipulating the internal hyperfine state of a qubit (qubit state) may be referred to as a “composite pulse” or simply a “pulse”, and the resulting time-dependent pattern of the two-photon Rabi frequency Ω(t) may be referred to as the “amplitude” of the pulse or simply a “pulse”, which are illustrated and further described below. The detuning frequency δ=ω ₁ −ω ₂ −ω ₀₁ may be referred to as the detuning frequency of the composite pulse or the detuning frequency of the pulse. The amplitude of the two-photon Rabi frequency Ω(t), determined by the amplitudes of the first and second laser beams, may be referred to as the “amplitude” of the composite pulse.

It should be noted that the specific atomic species used in the description provided herein are merely examples of atomic species that have a stable and well-defined two-level energy structure and optically accessible excited states when ionized, and are not intended to limit the possible configurations, specifications, etc. of the trapped ion quantum computer according to the present disclosure. For example, other ion species include alkaline earth metal ions (Be ⁺ , Ca ⁺ , Sr ⁺ , Mg ⁺ , and Ba ⁺ ) or transition metal ions (Zn ⁺ , Hg ⁺ , Cd ⁺ ).

FIG. 5 is provided to help visualize the ion's qubit state, represented as a point on the surface of a Bloch ball 500 with azimuthal angle φ and polar angle θ. As described above, application of a composite pulse produces Rabi oscillations between qubit states |0〉 (represented as the north pole of the Bloch ball) and |1〉 (represented as the south pole of the Bloch ball). Adjusting the duration and amplitude of the composite pulse flips the qubit state from |0〉 to |1〉 (i.e., from the north pole to the south pole of the Bloch ball) or flips the qubit state from |1〉 to |0〉 (i.e., from the south pole to the north pole of the Bloch ball). This application of a composite pulse is referred to as a "π pulse." Furthermore, by adjusting the duration and amplitude of the composite pulse, the qubit state |0> can be transformed into a superposition state |0>+|1>, where the two qubit states |0> and |1> are added and equally weighted in phase (normalization factors of the superposition states are omitted below for convenience), and the qubit state |1> can be transformed into a superposition state |0>-|1>, where the two qubit states |0> and |1> are added and equally weighted but out of phase. This application of the composite pulse is called a "π/2 pulse". More generally, the superposition of the two added and equally weighted qubit states |0> and |1> is represented by a point on the equator of the Bloch sphere. For example, the superposition states |0>±|1> correspond to points on the equator with azimuth angles φ of zero and π, respectively. The superposition state corresponding to a point on the equator at azimuth angle φ is expressed as |0〉+e ^iφ |1〉 (e.g., |0〉±i|1〉 for φ=±π/2). Transformation between two points on the equator (i.e., rotation around the Z axis of the Bloch sphere) can be implemented by shifting the phase of the composite pulse.

In a trapped ion quantum computer, the motional mode can act as a data bus mediating the entanglement between two qubits, which is used to perform XX gate operations. That is, each of the two qubits is entangled with a motional mode, and the entanglement is transferred to the entanglement between the two qubits by using motional sideband excitation, as described below. Figures 6A and 6B show schematic diagrams of the motional sideband spectrum of an ion in the chain 102 at a motional mode |n _ph > _p with a mode frequency ω _p , according to one embodiment. As shown in Figure 6B, when the detuning frequency of the composite pulse is zero (i.e., when the frequency difference between the first and second laser beams is tuned to the carrier frequency δ = ω ₁ - ω ₂ - ω ₀₁ = 0), a simple Rabi oscillation (carrier transition) occurs between the qubit states |0 > and |1 >. When the detuning frequency of the composite pulse is positive (i.e., the frequency difference between the first and second laser beams is tuned higher than the carrier frequency, δ = ω ₁ - ω ₂ - ω ₀₁ = μ〉0, called the "blue sideband"), Rabi oscillations occur between the combined qubit motional states |0〉│n _ph 〉 _p and |1〉│n _ph +1〉 _p (i.e., when the qubit state |0〉 flips to |1〉, a transition occurs from the pth motional mode with n-phonon excitation, represented by |n _ph 〉 _p , to the pth motional mode with (n _ph +1)-phonon excitation, represented by |n _ph +1〉 _p ). If the detuning frequency of the composite pulse is negative (i.e., the frequency difference between the first and second laser beams is tuned lower than the carrier frequency by the mode frequency ω _p │n _ph 〉 _p , δ = ω ₁ - ω ₂ - ω ₀₁ = -μ<0, called the "red sideband"), Rabi oscillations between the combined qubit motional states │0〉│n _ph 〉 _p and │1〉│n _ph -1〉 _p occur (i.e., when flipping from qubit state │0〉 to │1〉, a transition occurs from the motional mode │n _ph 〉 _p to the motional mode │n _ph -1〉 _p with one less phonon excitation). A π/2 pulse in the blue sideband applied to the qubit transforms the combined qubit motional state │0〉│n _ph 〉 _p into a superposition of │0〉│n _ph 〉 _p and │1〉│n _ph +1〉 _p . A π/2 pulse in the red sideband applied to a qubit transforms the combined qubit motional state |0〉|n _ph 〉 _p into a superposition of |0〉|n _ph 〉 _p and |1〉|n _ph -1〉 _p . If the two-photon Rabi frequency Ω(t) is small compared to the detuning frequency δ = ω ₁ - ω ₂ - ω ₀₁ = ±μ, the blue or red sideband transitions can be selectively driven. Thus, a qubit can be entangled in a desired motional mode by applying an appropriate type of pulse, such as a π/2 pulse, and then entangled with another qubit, resulting in entanglement between the two qubits. Entanglement between qubits is necessary to perform XX gate operations in a trapped ion quantum computer.

例えば、２つのキュービット（ｉ番目とｊ番目のキュービット）が両方とも最初に超微細基底状態｜０〉（｜０〉_ｉ｜０〉_ｊで表される）にあり、その後、青側波帯のπ／２パルスがｉ番目のキュービットに適用される場合、ｉ番目のキュービットと運動モード｜０〉_ｉ｜ｎ_ｐｈ〉_ｐの組み合わせ状態は、｜０〉_ｉ｜ｎ_ｐｈ〉_ｐと｜１〉_ｉ｜ｎ_ｐｈ＋１〉_ｐの重ね合わせに変換されるため、２つのキュービットと運動モードの組み合わせ状態は、｜０〉_ｉ｜０〉_ｊ｜ｎ_ｐｈ〉_ｐと｜１〉_ｉ｜０〉_ｊ｜ｎ_ｐｈ＋１〉_ｐの重ね合わせに変換される。赤側波帯のπ／２パルスがｊ番目のキュービットに適用される場合、ｊ番目のキュービットと運動モード｜０〉_ｊ｜ｎ_ｐｈ〉_ｐの組み合わせ状態は、｜０〉_ｊ｜ｎ_ｐｈ〉_ｐと｜１〉_ｊ｜ｎ_ｐｈ－１〉_ｐの重ね合わせに変換されるため、組み合わせ状態｜０〉_ｊ｜ｎ_ｐｈ＋１〉_ｐは、｜０〉_ｊ｜ｎ_ｐｈ＋１〉_ｐと｜１〉_ｊ｜ｎ_ｐｈ〉_ｐの重ね合わせに変換される。 As mentioned above, an XX gate operation can be performed on two qubits (i and j qubits) by controlling and/or directing the transformation of the combined qubit motional states. In general, an XX gate operation (with maximal entanglement) transforms the two-qubit states |0〉 _i |0〉 _j , |0〉 _i |1〉 _j , |1〉 _i |0〉 _j and |1〉 _i |1〉 _j , respectively, as

For example, if two qubits (i-th and j-th qubits) are both initially in the hyperfine ground state |0〉 (denoted by |0〉 _i |0〉 _j ) and then a π/2 pulse in the blue sideband is applied to the i-th qubit, the combined state of the i-th qubit and the kinetic mode |0〉 _i | _nph 〉 _p is transformed into a superposition of |0〉 _i | _nph 〉 _p and |1〉 _i | _nph +1〉 _p , and so the combined state of the two qubits and the kinetic mode is transformed into a superposition of |0〉 _i |0〉 _j | _nph 〉 _p and |1〉 _i |0〉 _j | _nph +1〉 _p . When a π/2 pulse in the red sideband is applied to the jth qubit, the combination state of the jth qubit and motional mode |0〉 _j | _nph 〉 _p is transformed into a superposition of |0〉 _j | _nph 〉 _p and |1〉 _j | _nph -1〉 _p , and so the combination state |0〉 _j | _nph +1〉 _p is transformed into a superposition of |0〉 _j | _nph +1〉 _p and |1〉 _j | _nph 〉 _p .

Thus, by applying a π/2 pulse in the blue sideband to the i qubit and a π/2 pulse in the red sideband to the j qubit, the combined state of the two qubits and motional modes |0〉 _i |0〉 _j |n _ph 〉 _p can be transformed into a superposition of |0〉 _i |0〉 _j |n _ph 〉 _p and |1〉 _i |1〉 _j |n _ph 〉 _p , and the two qubits are now in an entangled state. As will be apparent to those skilled in the art, two qubit states entangled with motional modes having a different number of phonon excitations than the initial number of phonon excitations n _ph (i.e., |1> _i |0> _j |n _ph +1> _p and |0> _i |1> _j |n _ph -1> _p ) can be removed by a sufficiently complex pulse sequence, so that the combined state of the two qubits and motional modes after the XX gate operation may be considered disentangled because the initial number of phonon excitations n _ph in the pth motional mode does not change at the end of the XX gate operation. Thus, the qubit states before and after the XX gate operation are generally described below without including motional modes.

ここで、

であり、η_ｐ ^ｉ、ｊは、ｉ番目（ｊ番目）のイオンとモード周波数ω_ｐを有するｐ番目の運動モードの間の結合強度を定量化するラムディッケ（Ｌａｍｂ－Ｄｉｃｋｅ）パラメータであり、ｇ（ｔ）は、ｇ（ｔ）＝Ω（ｔ）ｓｉｎ（ψ（ｔ））と定義されるパルス関数であり、ψ（ｔ）はパルスの累積位相関数（単に「位相関数」とも呼ばれる）

であり、ψ_０は、一般性を失うことなく、簡単にするために以下ゼロ（０）と見なすことができる初期位相であり、Ｐは運動モードの数（鎖１０２内のイオンの数Ｎに等しい）である。 More generally, the combined state of the i-th and j-th qubits, transformed by applying a composite pulse of sidebands for duration τ (called the “gate duration”) and with amplitude function Ω(t) and detuning frequency function μ(t), can be written in terms of the entanglement interaction χ _i,j (τ) as follows:

here,

where η _p ^i,j is the Lamb-Dicke parameter that quantifies the coupling strength between the ith (jth) ion and the pth motional mode with mode frequency ω _p , g(t) is the pulse function defined as g(t) = Ω(t) sin(ψ(t)), and ψ(t) is the cumulative phase function of the pulse (also called simply the “phase function”).

where ψ ₀ is the initial phase, which, without loss of generality, for simplicity can be considered as zero (0) hereafter, and P is the number of motional modes (equal to the number N of ions in the chain 102).

Constructing a Pulse for Entanglement Gate Operations Using the entanglement between two qubits (trapped ions) described above, an XX gate operation can be performed. The XX gate operations (XX gates), together with the single qubit operations (R gates), form a set of universal gates {R, XX} that can be used to construct a quantum computer to perform a desired computational process. In constructing a pulse for delivery to a chain of trapped ions 102 to perform an XX gate operation between two trapped ions (e.g., the i-th and j-th trapped ions) in the chain 102, the amplitude function Ω(t) and detuning frequency function μ(t) of the pulse are adjusted as control parameters to ensure that the pulse performs the desired XX gate operation by imposing the following gate requirements:

Gate requirement 1: First, to perform an XX gate operation XX(θ _ij ) with a desired rotation angle θ _ij (0<θ _ij ≦π/2), the entanglement interaction χ _ij generated by the pulse between the ith and jth trapped ions must be equal to the rotation angle θ _ij /4. A fully entangled XX gate requires θ _ij =π/2. This first requirement is also called the requirement for non-zero entanglement interactions.

Gate Requirement 2: Second, we can minimize the laser power required to implement the pulse so that the constructed pulse is of optimal power. This second requirement is also called the requirement that the laser pulse have a minimum peak power.

Gate Requirement 3: Third, all trapped ions in chain 102 that are moving from their initial position when a motional mode is excited by the delivery of a pulse must return to their initial position at the end of the XX gate operation. This third requirement is also called the ion mode decoupling requirement, where trapped ions excited by a motional mode return to their original position and momentum value.

However, the train of laser pulses constructed as above to perform computational processes in large-scale quantum computing raises complex technical problems with respect to the optics and electronics used to modulate such laser pulses. In the embodiments described herein, a degree of freedom in the design and manufacture of an ion trap for confining ions, such as ion trap 102, is newly introduced to compensate for some of the added technical complexity in optics and electronics. It is therefore desirable to design and manufacture an ion trap that modulates the confinement potential of the ion trap to configure the mode frequency ω _p such that the constructed pulses are simplified. In some embodiments, the pulses are selected to have a certain symmetry, such as the phase of the pulses being reversed at the midpoint of the gate duration of the pulses to make the gate operation more robust. Thus, the following configuration conditions are imposed when constructing the pulses used to perform the XX gate operation:

Condition 1: The motional mode structure is modulated so that the mode frequency ω _p is an integer multiple of 4π/τ with a small error δk _p , where τ is the gate duration and k _p is a positive integer (i.e., ω _p = 4(k _p + δk _p )π/τ).

Condition 2: The pulse function g(t) is chosen to have the symmetry g(t+π/2)=-g(t) for t∈[0,π/2]. That is, the pulse is a combination of two consecutive pulse segments of opposite phase but the same shape (i.e., the same detuned frequency μ(t) and the same amplitude function Ω(t)).

Condition 3: In one example, the pulse can be further simplified by requiring that the pulse amplitude function Ω(t) be a constant Ω for the gate duration τ, and the pulse detuning frequency function μ(t) be an integer multiple of 2π/τ, μ(t) = 2lπ/τ, where l is a positive integer. That is, the pulse function is written as g(t) = Ω sin(2lπt/τ) for t ∈ [0,π/2]. Thus, the positive integer l that determines the constant amplitude Ω of the pulse and the detuning frequency μ of the pulse are control parameters that are adjusted in constructing the pulses used to perform the XX gate operation.

As mentioned above, gate requirement 1 (also called the non-zero entanglement interaction requirement) requires that the entanglement interaction χ _ij generated between the i-th and j-th trapped ions by the pulse has a rotation angle value θ _ij (0<θ _ij ≦π/2).

として簡略化することができ、ここで、ｐ_ｌは２ｋ_ｐｌ＝ｌを満たす。 When conditions 1, 2, and 3 are imposed, the entanglement interactions χ _ij are

where p _l satisfies 2k _{p l} =l.

Gate requirement 2 (also called the requirement that the laser pulse have a minimum peak power) requires that for a given set of Lambdicke parameters η _p ^i,j and mode frequencies ω _p , a positive integer l is chosen such that the constant amplitude Ω is the lowest value to satisfy the requirement of non-zero entanglement interactions. That is, a positive integer l is chosen such that the gate power ratio, defined by the dimensionless quantity |χ _ij /Ω ² τ ² |, is maximized. Once a positive integer l is chosen, the constant amplitude Ω can be calculated based on gate requirement 1, χ _ij =θ _ij /4.

として制限され、ここで、φ_ｐは、時間ｔ＝０でのｐ番目の運動モードに関連する初期位相であり、Ｔ_ｐは、ｐ番目の運動モードの温度であり、ｈは、換算プランク定数であり、ｋ_Ｂは、ボルツマン定数である。 Gate requirement 3 (also called the ion mode decoupling requirement) requires that trapped ions excited by the delivery of a pulse return to their initial position. After application of a pulse for gate duration τ, the residual coupling α of the pth motional mode with the ith and jth trapped ions, determined by the Lamb-Dicke parameters η _p ^i,j , the pulse function g(t) of the pulse, and the mode frequency ω _p of the pth motional mode, is required to be smaller than an error budget of ε, such that the displacement of the ith and jth trapped ions from their initial positions in phase space is

where φ _p is the initial phase associated with the pth motional mode at time t=0, T _p is the temperature of the pth motional mode, h is the reduced Planck constant, and k _B is the Boltzmann constant.

として簡略化することができる。 When conditions 1, 2, and 3 are imposed, the residual coupling α is

This can be simplified as:

であるので、

に制限される。 Assuming that the average phonon number _n is less than one ( _n < 1) by known laser cooling techniques, e.g., Doppler cooling or resolved sideband cooling, and that the motional modes have been sufficiently cooled such that the initial phase _φ associated with the pth motional mode is zero, for simplicity without loss of generality, the residual coupling α can be calculated as

Therefore,

is limited to.

のような誤差δｋ_ｐの異なる次数の項の級数（ｓｅｒｉｅｓ ｏｆ ｔｅｒｍｓ）として展開することができ、ここで、

である。 In the range where the error δk _p of the motion frequency ω _p is small (i.e., |δk _p |<<1), the upper limit of the residual coupling α is

The error _δk can be expanded as a series of terms of different orders such as:

It is.

Note that for a given error budget ε in the residual coupling α, if no motional mode p exists that satisfies _kp = l/2, then the bounds of the required error _δkp at the mode frequency _ωp are much less stringent for pulses with a positive integer l even than for pulses with a positive integer l odd. Thus, in some embodiments, the positive integer l is selected to be an even number.

である。条件２のようなパルスの位相に反転させ、傍観イオンの位相を変化しないままにしておくことによって、クロストークは抑制される。 It should also be noted that pulses constructed in the manner described herein can be easily adapted to suppress crosstalk errors incurred by spectator ions in the vicinity of the ion intended for manipulation by XX gate operations. An XX gate, XX(θ _ij ), can be implemented as a combination of two XX gates, XX(θ _ij/2 ), and a single qubit gate by pulses that satisfy condition 2. Due to conditions 1 and 2, the entanglement rotation angle accumulated in the first half of the gate duration τ is already the same as that accumulated in the second half of the gate duration τ. That is,

By inverting the phase of the pulse as in condition 2 and leaving the phase of the spectator ions unchanged, crosstalk is suppressed.

Mode Configuration In the embodiments described herein, the ion trap for confining ions is operated to satisfy Condition 1 (i.e., the mode frequency _ωp is an integer multiple of 4π/τ with a small error _δkp , _ωp = 4( _kp + _δkp )π/τ, where _kp is a positive integer and τ is the gate duration). That is, the static (DC) voltage _Vs and the radio frequency (RF) voltages applied to the control electrodes of the ion trap 200 to provide the confining potentials are adjusted such that the resulting motional mode structure satisfies Condition 1. FIG. 7 shows numerically calculated profiles of the static (DC) voltages Vs applied to several electrodes of an exemplary Sandia High Optical Access (HOA) 2.0 trap 700 confining three ions 702, 704 and 706 such that the resulting motional mode structure satisfies Condition ₁ . In this example ion trap 700, the pulse construct to perform the XX gate operation is selected to have a gate duration τ=69.466 μs, and the three motional modes, center of mass (COM), tilt, and zigzag, have mode frequencies ω _p corresponding to positive integers k _p =97, 95, and 92, respectively. In this example HOA2.0 trap 700, a radio frequency (RF) voltage at a frequency of 50.6 MHz is further applied to the trapping electrodes with an amplitude of 289.71 V. The profile of the static (DC) voltage V _S shown in Figure 7 traps three ions 702, 704, and 706 linearly aligned in a chain with a spacing of about 4.3 μm on the surface of the ion trap 700 at a distance of about 71 μm. The static (DC) voltage V _S creates slightly higher radial confinement on the central ion 704 than on the end ions 702 and 704, generating mode frequencies ω _p of 2.793×2π MHz for the COM mode, 2.735×2π MHz for the tilt mode, and 2.649×2π MHz for the zigzag mode. These mode frequencies ω _p deviate slightly from the mode frequencies ω _p corresponding to the positive integers k _p =97, 95, and 92, respectively, that satisfy Condition 1. However, the effect of this deviation (characterized by an error δk _p ) is small, as explained below.

Figure 8 shows the numerically calculated gate power ratio |χ ij /Ω 2 τ 2 | 800 for the entanglement interaction between ions 702 and 704 as a function of the positive integer l that determines the detuning frequency μ as μ=2lπ/τ in the example shown in Figure 7. The gate power ratio |χ _ij /Ω ² τ ² | is maximized (Gate Requirement ² ) if the positive integer l is chosen to be 192 if it is even and 193 if it is odd. The l=192 pulse requires slightly more power at _Ω / ^2π =0.133 MHz compared to 0.101 MHz for the l=193 pulse.

9A and 9B show pulse functions g(t) 902 and 904 calculated numerically for the example shown in FIG. 7 by choosing positive integers l of 193 and 192, respectively. The pulse function g(t) 904 with l=192 in FIG. 9B exhibits a cusp at the midpoint of the gate duration τ=69.466 μs, which is not present in the pulse function g(t) 902 with l=193 in FIG. 9A. However, this cusp of the pulse function g(t) 904 with l=192 can be effectively removed by applying an appropriate single-qubit gate at the point of the cusp.

FIG. 10 shows the numerically calculated residual coupling α as a function of the variation δω of the mode frequency _ωp for the example shown in FIG. 7. For simplicity, all three motional modes are chosen to have the same frequency variation δω. Both the residual coupling α 1002 for l=193 and the residual coupling α 1004 for l=192 contain non-zero values at the frequency variation δω=0 only due to static errors in the motional mode configuration (i.e., deviations of the mode frequency _ωp from the value of the mode frequency _ωp that exactly satisfies condition 1), while larger non-zero values at the frequency variation δω increase due to hardware errors or noise. The residual coupling α 1004 for ^{l=192 (even) is much smaller than the residual coupling α 1002 for l=193 (odd) due to the δkp2} _dependence ^of the residual coupling α for even l instead of the _δkp4 dependence for odd l, as described above. Note that this advantage of even l over odd l exists when l ≠ _2kp for any motion mode, as is the case in the examples described herein. The tradeoff is that a pulse with l = 192 requires approximately 30% more power compared to a pulse with l = 193. The choice between even and odd l must be made on a case-by-case basis, taking into due consideration hardware limitations.

Figure 11 shows a flowchart illustrating a method 1100 for performing a computation using a quantum computer.

In block 1110, the motional mode structure of the plurality of trapped ions is modulated. Each of the plurality of trapped ions has two frequency separated states that define a qubit.

In block 1120, the detuning frequency and amplitude functions of the laser pulse are used to calculate entanglement interactions between pairs of trapped ions among the plurality of trapped ions.

In block 1130, the quantum computation is performed in the quantum computer by applying a laser pulse having the calculated detuning frequency function and amplitude function to the pair of trapped ions for the gate duration.

As described above, when generating a pulse to perform an entanglement gate operation between two trapped ions in a chain, the control parameters (pulse detuning frequency function and amplitude function) are determined to satisfy the gate requirements. In doing so, the constructed pulses are simple and the motional mode structure of the trapped ions in the chain is configured to be practically implementable in speed and bandwidth limited hardware. When designing an ion trap to confine ions and modulate the motional mode structure in a desired manner, the electrodes of the ion trap can be structured to provide a desired confinement potential. With appropriate technological advances, an increased number of electrodes and various shapes of electrodes can be utilized to induce the desired confinement potential. The embodiments described herein provide a way to reallocate the technical complexities that must be overcome to build a practical large-scale quantum computer.

While the forgoing is directed to specific embodiments, other and further embodiments may be devised without departing from the basic scope thereof, which is determined by the following claims.

Claims (20)

1. A method of performing a computation using a quantum computer, comprising:
modulating a motional mode structure of a plurality of trapped ions, each of the plurality of trapped ions having two frequency separated states that define a qubit;
calculating a detuning frequency function and an amplitude function of a laser pulse, the detuning frequency function and the amplitude function causing an entanglement interaction between pairs of trapped ions of the plurality of trapped ions;
and performing a quantum computation on the quantum computer by applying a laser pulse having the calculated detuning frequency function and the amplitude function to the pair of trapped ions for a gate duration. The method of claim 1, wherein modulating the kinetic mode structure includes adjusting a confinement potential of an ion trap that traps the plurality of trapped ions such that a kinetic mode frequency is an integer multiple of 4π divided by the gate duration. The method of claim 1, wherein the laser pulse comprises two successive pulse segments of the same detuned frequency function and the same amplitude function, but in antiphase. the amplitude function of the laser pulse is constant during the gate duration;
2. The method of claim 1, wherein the detuning frequency function of the laser pulse is an integer multiple of 2π divided by the gate duration. The method of claim 1, wherein the step of calculating the detuning frequency function and the amplitude function of the laser pulse is based on a first gate requirement of non-zero entanglement interactions. The method of claim 5, wherein the step of calculating the detuning frequency function and the amplitude function of the laser pulse is based on a second gate requirement that the laser pulse have a minimum peak power. The method of claim 6, wherein the step of calculating the detuning frequency function and the amplitude function of the laser pulse is based on a third gate requirement of ion mode decoupling for the plurality of trapped ions. A non-volatile computer readable medium containing computer program instructions that, when executed by a processor, cause the processor to:
calculating a detuning frequency function and an amplitude function of the laser pulse to induce entanglement interactions between pairs of trapped ions of the plurality of trapped ions;
performing a quantum computation in a quantum computer by applying a laser pulse having the calculated detuning frequency function and the calculated amplitude function to the pair of trapped ions for a gate duration;
A non-volatile computer readable medium, wherein motional mode frequencies of the plurality of trapped ions are modulated to be integer multiples of 4π divided by the gate duration. The non-volatile computer readable medium of claim 8, wherein the laser pulse comprises two successive pulse segments of the same detuned frequency function and the same amplitude function, but in antiphase. the amplitude function of the laser pulse is constant during the gate duration;
9. The non-volatile computer-readable medium of claim 8, wherein the detuning frequency function of the laser pulse is an integer multiple of 2π divided by the gate duration. The non-volatile computer-readable medium of claim 8, wherein the calculation of the detuning frequency function and the amplitude function of the laser pulse is based on a first gate requirement of non-zero entanglement interactions. The non-volatile computer readable medium of claim 11, wherein the calculation of the detuning frequency function and the amplitude function of the laser pulse is based on a second gate requirement where the laser pulse has a minimum peak power. The non-volatile computer readable medium of claim 12, wherein the calculation of the detuning frequency function and the amplitude function of the laser pulse is based on a third gate requirement of ion mode decoupling for the plurality of trapped ions. a plurality of trapped ions in an ion trap, each of the trapped ions having two hyperfine states that define a qubit;
a controller having a non-volatile memory having a number of instructions stored therein;
The instructions, when executed by a processor, cause the quantum computing system to:
calculating a detuning frequency function and an amplitude function of a laser pulse to induce entanglement interactions between pairs of trapped ions of the plurality of trapped ions;
performing a quantum computation in a quantum computer by applying a laser pulse having the calculated detuning frequency function and amplitude function to the pair of trapped ions for a gate duration;
A quantum computing system, wherein a confining potential of the ion trap is modulated such that a motional mode frequency of the plurality of trapped ions is an integer multiple of 4π divided by the gate duration. The quantum computing system of claim 14, wherein each of the trapped ions is an ion having a nuclear spin and an electron spin such that the difference between the nuclear spin and the electron spin is zero. The quantum computing system of claim 14, wherein the laser pulse includes two successive pulse segments that are the same detuning frequency function and the same amplitude function, but in antiphase. the amplitude function of the laser pulse is constant during the gate duration;
15. The quantum computing system of claim 14, wherein the detuning frequency function of the laser pulse is an integer multiple of 2π divided by the gate duration. The quantum computing system of claim 14, wherein the calculation of the detuning frequency function and the amplitude function of the laser pulse is based on a first gate requirement of non-zero entanglement interactions. The quantum computing system of claim 18, wherein the calculation of the detuning frequency function and the amplitude function of the laser pulse is based on a second gate requirement that the laser pulse have a minimum peak power. 20. The quantum computing system of claim 19, wherein calculation of the detuning frequency function and the amplitude function of the laser pulse is based on a third gate requirement of ion mode decoupling for the plurality of trapped ions.

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